# ✅ Arc Length Formula ⭐️⭐️⭐️⭐️⭐

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## Arc Length

Arc length is better defined as the distance along the part of the circumference of any circle or any curve (arc). Any distance along the curved line that makes up the arc is known as the arc length. A part of a curve or a part of a circumference of a circle is called Arc. All of them have a curve in their shape. The length of an arc is longer than any straight line distance between its endpoints (a chord).

## What is Arc Length?

The arc length is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the end-points of the arc. For example, in the circle shown below, OP is the arc of the circle with center Q. The arc length of this arc OP is given as L.

## Arc Length Formula

The length of an arc can be calculated using different formulas, based on the unit of the central angle of the arc. The measurements of the central angle can be given in degrees or radians, and accordingly, we calculate the arc length of a circle. For a circle, the arc length formula is θ times the radius of a circle.

The arc length formula in radians can be expressed as, arc length = θ × r, when θ is in radian. Arc Length = θ × (π/180) × r, where θ is in degree, where,

• L = Length of an Arc
• θ = Central angle of Arc
• r = Radius of the circle

### Arc Length Formula in Radians

The arc length of a circle can be calculated using different formulas, based on the unit of the center angle of the arc. The arc length formula in radians can be expressed as,

Arc Length = θ × r

where,

• L = Arc Length
• θ = Center angle of the arc in radians
• r = Radius of the circle

### Denotations in the Arc Length Formula

• s is the arc length
• r is the radius of the circle
• θ is the central angle of the arc

### How to Find Arc Length of a Curve?

The arc length of an arc of a circle can be calculated using different methods and formulas based on the given data. Some important cases are given below,

• find arc length with the radius and central angle
• find arc length without the radius
• find arc length without the central angle

### How to Find Arc Length With the Radius and Central Angle?

The arc length of a circle can be calculated with the radius and central angle using the arc length formula,

• Length of an Arc = θ × r, where θ is in radian.
• Length of an Arc = θ × (π/180) × r, where θ is in degree.

### How to Find Arc Length Without the Radius?

The arc length of a circle can be calculated without the radius using:

Central angle and the sector area:

• Multiply the sector area by 2 and further, divide the result by the central angle in radians.
• Find the square root of the result of the division.
• Multiply this obtained root by the central angle again to get the arc length.
• The units of this calculated arc length will be the square root of the sector area units.

Example: Calculate the arc length of a curve with sector area 25 square units and the central angle as 2 radians.

We have,

Sector area = 25 units

• Step 1: Sector area × 2 = 25 × 2 = 50
• Step 2: 50/central angle = 50/2 = 25
• Step 3: √25 = 5
• Step 4: 5 × central angle = 5 × 2 = 10 units

Thus, arc length = 10 units

Central angle and the chord length:

• Divide the central angle in radians by 2 and further, perform the sine function on it.
• Divide the given chord length by twice the result of step 1. This calculation gives you the radius as result.
• Multiply the radius by the central angle to get the arc length.

Example: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The central angle subtended by the arc is 2 radians.

We have,

Chord length = 5 units

• Step 1: Central angle/2 = 2/2 = 1
• Step 2: Sin(1) = 0.841
• Step 3:Chord length/ (2 × 0.841) = 5/ 1.682 = 2.973 units = radius
• Step 4: Arc length = radius × central angle = 2.973 × 2 = 5.946 units

Thus, arc length = 5.946 units

### How to Find Arc Length Without the Central Angle?

The arc length of a circle can be calculated without the angle using:

• Multiply the sector area by 2.
• Then divide the result by the radius squared (the units should be the same) to get the central angle in radians.
• Multiply the central angle by the radius to get the arc length.

Example: Calculate the arc length of a curve with sector area 25 square units and radius as 2 units.

We have,

Sector area = 25 units

Central angle = 2 units

• Step 1: Sector area × 2 = 25 × 2 = 50
• Step 3: Arc length = radius × central angle = 2 × 12.5 = 25 units

Thus, arc length = 25 units

• Divide the chord length by twice the given radius.
• Find the inverse sine of the obtained result.
• Double the result of the inverse sine to get the central angle in radians.
• Multiply the central angle by the radius to get the arc length.

Example: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The radius of the circle is 2 units.

We have,

Chord length = 5 units

Central angle = 2 units

• Step 1: Chord length/(2 × radius) = 5/(2 × 2) = 1.25
• Step 2: Sin-1(1.25) = 0.949
• Step 3: Central angle = 2 × 0.949 = 1.898 radians
• Step 4: Arc length = radius × central angle = 2 × 1.898 = 3.796 units

Thus, arc length = 3.796 units

☛ Important Notes on Arc Length

Given below are key highlights on the concept of arc length.

• Arc Length = θ × r, where θ is in radian.
• Arc Length = θ × (π/180) × r, where θ is in degree.

### Example Questions Using the Formula for Arc Length

Question 1: Calculate the length of an arc if the radius of an arc is 8 cm and the central angle is 40°.

Solution:

Central angle, θ = 40°

Arc length = 2 π r ×  (θ/360°)

So, s = 2 × π × 8 × (40°/360°)

= 5.582 cm

Question 2: What is the arc length for a function f(x) = 6 between x = 4 and x = 6?

Solution:

Since the function is a constant, the differential of it will be 0. So, the arc length will now be-

So, arc length (s) = (6 – 4) = 2.

### Practice Questions Based on Arc Length Formula

1. What would be the length of the arc formed by 75° of a circle having the diameter of 18 cm?
2. The length of an arc formed by 60° of a circle of radius “r” is 8.37 cm. Find the radius (r) of that circle.
3. Calculate the perimeter of a semicircle of radius 1. cm using the arc length formula.

### Arc Length Calculator

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area.

The length of an arc depends on the radius of a circle and the central angle `θ`. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that:`L / θ = C / 2π`

As circumference `C = 2πr`,`L / θ = 2πr / 2π``L / θ = r`

We find out the arc length formula when multiplying this equation by θ:`L = r * θ`

Hence, the arc length is equal to radius multiplied by the central angle (in radians).

### Area of a sector of a circle

We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions,`A / θ = πr² / 2π``A / θ = r² / 2`

The formula for the area of a sector is:`A = r² * θ / 2`

### How to find the length of an arc and sector area: an example

1. Decide on the radius of your circle. For example, it can be equal to 15 cm. (You can also input the diameter into the arc length calculator instead.)
2. What will be the angle between the ends of the arc? Let’s say it is equal to 45 degrees, or π/4.
3. Calculate the arc length according to the formula above: `L = r * θ = 15 * π/4 = 11.78 cm`.
4. Calculate the area of a sector: `A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm²`.
5. You can also use the arc length calculator to find the central angle or the circle’s radius. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.

### Calculus II – Arc Length

Math Formulas ⭐️⭐️⭐️⭐️⭐